Properties

Label 1764.cz
Modulus $1764$
Conductor $1764$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,35,22]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(95,1764))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1764\)
Conductor: \(1764\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.84986492613867422664202192058457487324646368884457377881199611944629462212222730322047304527187202914910208.2

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(37\)
\(\chi_{1764}(95,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{16}{21}\right)\)
\(\chi_{1764}(191,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{21}\right)\)
\(\chi_{1764}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{1764}(443,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{21}\right)\)
\(\chi_{1764}(599,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{10}{21}\right)\)
\(\chi_{1764}(695,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{21}\right)\)
\(\chi_{1764}(947,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{21}\right)\)
\(\chi_{1764}(1103,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{4}{21}\right)\)
\(\chi_{1764}(1199,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{1764}(1355,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{21}\right)\)
\(\chi_{1764}(1607,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{21}\right)\)
\(\chi_{1764}(1703,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{21}\right)\)